Let \(\Gamma ^{\prime }:=\operatorname{Gal}(\mathbb {Q}(\mu _{p^{\infty }})/\mathbb {Q})\simeq \mathbb {Z}_p^{\times }\)  and \(\Delta :=\ker \left(\Gamma ^{\prime }\rightarrow \Gamma \right)\)  so that \(\Delta \)  can be identified with \(\mu _{p-1}\subset \mathbb {Z}_p^{\times }\) . For \(i=1,2\)  as before, set \(\chi _t(\Gamma ^{\prime }, E_i):=\chi _t(\Gamma ^{\prime }, \operatorname{Sel}(E_i/\mathbb {Q}(\mu _{p^{\infty }})))\) . Let \(E\)  be an elliptic curve with good ordinary reduction at \(p\)  and \(\tilde{E}\)  be the reduced curve at \(p\) . The extensions \(\mathbb {Q}^{cyc}\)  and \(\mathbb {Q}(\mu _{p^{\infty }})\)  are deeply ramified extensions (see [1]}). Suppose \(F_{\infty }/\mathbb {Q}\)  is a deeply ramified extension. Let \(l\)  be a prime and \(w|l\)  a prime of \(F_{\infty }\)  above \(l\) . If \(l\ne p\) , there is a canonical isomorphism
\(H^1(F_{\infty ,w}, E)_{p^{\infty }}\simeq H^1(F_{\infty ,w}, E_{p^{\infty }}).\) 
